#104699

order	512
nilpotency class	3
isoclinism	isoclinic class of #104645
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

535 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,4,6)(3,5,7,8)(10,14)(11,15)(12,16)(17,18)(19,26)(20,30)(21,24)(22,25)(23,27)(28,31)(29,32);(5,8)(9,10,13,11)(12,15,16,14)(17,19,22,28)(18,23,25,32)(20,29,21,27)(24,31,30,26);(1,3)(2,5)(4,7)(6,8)(9,11,13,10)(12,14,16,15)(19,27)(20,21)(23,31)(24,30)(26,32)(28,29)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,d^c=dh,e^c=ei,f^c=fgi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11895);